Modules with RD-Composition Series over a Commutative Ring

Abstract

If R is a commutative ring,then we prove that every finitely generated R-module has a pure-composition series with indecomposable cyclic factors and any two such series are isomorphic if and only if R is a Bézout ring and a CF-ring. When R is a such ring,the length of a pure-composition series of a finitely generated R-module M is compared with its Goldie dimension and we prove that these numbers are equal if and only if M is a direct sum of cyclic modules. We also give an example of an artinian module over a noetherian domain,which has an RD-composition series with uniserial factors. Finally we prove that every pure-injective R-module is RD-injective if and only if R is an arithmetic ring.

Key Words:

• Pure-composition series
• RD-composition series
• Arithmetic ring
• Kaplansky ring
• Goldie dimension
• RD-injective module
• Pure-injective module

Acknowledgments